What does the statement "He wanted to show that fate ruled people's lives and that those who interfered with it did so to their own sorrow"...

The description of the fakir’s motives demonstrates that the Whites are soon going to be ruled by the fate of the monkey’s paw.

The Whites are curious about the monkey’s paw when their old friend the Sargent-Major brings it to them. However, they do not really take it seriously. They think it is a game, or just a story that their old friend is trying to entertain them with.

The Whites do seem curious about the history of the trinket. 

"And has anybody else wished?" persisted the old lady. "The first man had his three wishes.”

“Yes," was the reply; "I don't know what the first two were, but the third was for death. That's how I got the paw."

This is a warning that the paw is dangerous or evil. Why would a person wish for death after using it twice? Terrible things must have happened.

However, while they joke about it and use it to make a wish, they do not really feel that it will do anything. They do not destroy it, and they don’t listen to the warnings. Mr. White does get a little pensive and nervous, thinking he sees something in the fire.

It seems that with the Whites, the fakir proves his point. Their lives are apparently ruled by the fate of the paw. The wish for money results in their son’s death. That could be a coincidence, but in their grief they turn to the paw again.

The old man turned and regarded her, and his voice shook. "He has been dead ten days, and besides he—I would not tell you else, but—I could only recognize him by his clothing. If he was too terrible for you to see then, how now?"

When Mrs. White wishes for her son to be alive, and then they hear a knocking at the door, it is too much for Mr. White. He uses the third wish to get rid of the creature at the door. The Whites’ lives will never be the same, and all because they decided to play with fate.